5 Biomembranes

5.1 Motivation

Biological membranes are thin, flexible, selectively permeable surfaces that separatecells and their organelles from their environments. Biomembranes form closed struc-tures with an internal face oriented toward the interior of the compartment and anexternal face presented to the environment. Each type of cellular membrane has cer-tain distinctive activities determined largely by the unique set of proteins associatedwith that membrane. We can distinguish two different types of membrane proteins (i)integral proteins, which partly or entirely penetrate the membrane, and (ii) peripheralproteins, which are laterally attached to one side of the membrane. Biomembranes aretypically asymmetric, their interior and exterior faces can carry different proteins andhave different properties.Although biological membranes may have different properties and functions, all mem-branes share a common structural architecture that we will address in this chapter.They are rich in phospholipids, which spontaneously form a characteristic bilayerstructures in water. Membrane proteins and lipids can diffuse laterally or sidewaysthroughout the membrane, giving them their characteristic appearance of a fluid ratherthan a solid.

Figure 5.1: Electron microscopy of the cell membrane stained with osmium tetroxide illustrating the polarhead groups with a light 2nm space of hydrophobic tails sandwiched between them, adopted from [1]

Given the variable composition of cellular membranes, how can we be that the phos-pholipid bilayer structure is common to all biomembranes? Electron microscopy ofthin membrane sections provides the most direct evidence for the universality of thebilayer structure. Osmium tetroxide can be used for staining of phospholipids since itbinds strongly to the polar head groups. A cross section of a single membrane stainedwith osmium tetroxide looks like a railroad track. Figure 5.1 shows two thin dark lines,the stained head group complexes, and a uniform 2nm light space of the hydrophobictails sandwiched between them.

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5 Biomembranes

5.1.1 Lipid bilayers

In cell biology, the notion membrane is typically associated with the phospholipidbilayer and the proteins associated with it. In aqueous solutions, these proteins es-sentially display two kinds of non-covalent interactions which are referred to as hy-drophobic and hydrophilic. Long polymer molecules typically tend to adopt confir-mations, in which non-polar residues are predominantly sequestered such that theyavoid contact with water. The non-polar residues are said to be hydrophobic or water-avoiding. Polymer molecules favor confirmations, in which the polar head groupsare exposed to water. The polar head groups are referred to as being hydrophilic orwater-loving. A typical example is the arrangement of fatty acids at an oil water in-terface, where the hydrophilic polar heads would typcially be oriented towards thewater phase while the hydrophobic tails would be oriented towards the oil phase, seefigure 5.2.

hydrophilic headfatty acidhydrophobic tail

water

oil water interface

oilFigure 5.2: Oil water interface. Characteristic arrangement of fatty acid molecules with hydrophilic polarhead group oriented towards the water phase and hydrophobic tail oriented towards the oil phase

From an energetic point of view, lipid bilayers show an attractive arrangement sincethey display both hydrophobic and hydrophilic interactions. The nonpolar fatty acidchains of the phospholipid molecules are sequestered together away from the watersandwiched between the polar head groups to maximize hydrophobic interactions. Atthe same time, the ionic polar head groups are in direct contact with the aqueous phaseto maximize hydrophilic interactions. This dual nature of the molecules is referred toas amphiphilic.For energetic reasons, each lipid bilayer has an inherent optimal microstructure withand optimal spacing between the lipid molecules. Any perturbation to this optimal ar-rangement disturbes this energetically favorable microstructure. The lipid bilayer thusexhibits an inherent resistance to deformations that cause shape changes. Typical ex-amples are extension, for which the spacing between the head groups would increasethroughout the membrane, or bending for which the head group spacing would in-crease on the outside while it would decrease on the inside, see figure 5.3.One of the key issues of this chapter is the identification of characteristic macroscopicparamteres that display the nature of these intermolecular effects in a phenomenolog-ical way and account for the resistance of the cell membrane to extension, shear andbending. To this end, we will first look at a lipid bilayer structure that everybody caneasily reproduce and elaborate at home, the structure of soap bubbles. When havingunderstood how soap bubbles behave and how they can be described by mathemati-cal equations, we will turn to elaborating the structural behavior of the cell membranewhich is slightly more complicated but obey a similar set of equations from a mathe-matical point of view.

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5 Biomembranes

nxx nxx

nyy

nyy

nxy

nxy

nyx

nyx

myy myy

pz

Figure 5.3: Infinitesimal element of the cell membrane subject to tension causing in plane deformationand shear (left) and bending causing out of plane deformation (right)

5.1.2 Soap bubbles

Soap bubbles are fascinating structures that display many similar features as the cellmembrane. They can be used as model system to illustrate the qualitative behaviorof a lipid bilayer. Soap bubbles are an excellent example of a self-assembled system.Their surface consists of a thin layer of water trapped between two layers of surfactant,typically soap. The surfactant possesses hydrophilic heads attracted to the thin waterlayer. Its hydrophobic tails form the inner and outer surface of the bubble as sketchedfigure 5.4. When being disturbed, the bubble pops.

watersoap

air

soap bubble

airFigure 5.4: Lipid bilayer of soap bubbles - characteristic arrangement of soap molecules with a thinwater layer being sandwiched between the hydrophilic polar head groups while the hydrophobic tails areoriented to the non-polar air

The spherical shape of a soap bubble nicely displays the principle of energy mini-mization. Surface tension causes the bubble to form a sphere because this shape, asproposed by Archimedes and proven rigorously by Schwarz in 1884, is the minimalsurface enclosing a fixed given volume. The spherical shape can be visibly distortedby additional external forces, you can easily test this by blowing against the bubblesurface. If a bubble is subject to an enviroment without any additional external forcesacting on it, however, it should always remain nearly spherical as displayed in figure5.5.An interesting question to ask about soap bubbles is what is the radius r of a soap bub-ble that is blown up at a pressure ∆p? Here ∆p = pint − pout would be the pressuredifference between the inside and outside of the bubble. To answer this question, weconsider spherical soap bubble with initial radius r, that has a surface of A = 4 π r2

and a volume of V = 43 π r3. The inflation of the surface induces an internal energy

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5 Biomembranes

soap bubblesoap

water

Figure 5.5: Lipid bilayer of soap bubbles - characteristic arrangement of soap molecules with a thinwater layer being sandwiched between the hydrophilic polar head groups while the hydrophobic tails areoriented to the non-polar air

W int which is assumed to be proportional to the increase in membrane surface A. Inthe simplest case, W int = γ A where, for now, γ is introduces as a mere proportionalityconstant. Its unit is obvioulsy force per length and its physical interpretation will bediscussed later. The external work Wext is equal to the pressure difference ∆p acting onthe enclosed volume V, such that Wext = ∆p V. The total energy W of the bubble thusconsists of the internal energy W int and the external energy Wext.

W(r) = W int −Wext withW int = γ A = γ 4 π r2

Wext = ∆p V = ∆p 43 π r3

(5.1.1)

The minimum of the overall energy W with varying bubble radius r is obviously equiv-alent to the vanishing first variation δW with respect to r.

W(r)→ min δW(r) .= 0 withδW int = γ 8 π r

δWext = ∆p 4 π r2(5.1.2)

Evaluating the above equation γ 8 π r− ∆p 4 π r2 .= 0 we obtain the following simplerelation between the pressure difference ∆p and the bubble radius r

∆p = 2 γ1r

(5.1.3)

which has been developed independently by Young and Laplace more than 200 yearsago [7, 12]. In the literature, equation (5.1.3) is referred to as the Young-Laplace equa-tion. The historical controversy about its development is documented by Müller &Strehlow [8]. We will see later how this equation for spherical membranes such assoap bubbles can be derived in a more rigorous form.The cohesive forces between liquid molecules are responsible for the phenomenonwhich is referred to as surface tension. Cohesive forces between molecules are sharedbetween all neighboring molecules. Unlike molecules in the bulk of the liquid, moleculesclose to the surface are surrounded by neighboring molecules from only one side.These molecules on the surface thus exhibit stronger attractive forces upon their near-est neighbors than do those on the inside. This enhancement of intermolecular attrac-tive forces close to the surface is called surface tension, see figure 5.9.

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5 Biomembranes

air

air-water interface

water

no cohesive forces

cohesive forcesfrom all sides

Figure 5.6: Air water interface - molecular interpretation of surface tension

More than a century ago, an illustrative set of experiments on surface tension was car-ried out by Boys [2]. You can easily visualize the effect of surface tension by carefullylaying down a paper clip on a surface of water. Although the density of the paper clipshould be higher than that of water and you would expect it to sink down, it actuallyfloats on top of the water surface due to surface tension.

Surface tension Surface tension is typically measured in force per length related tothe units dynes per cm. Since 1 dyne = 10 mN, 1 dyne/cm = 1 mN/m. Alternatively, es-pecially in thermodynamics, the notion surface energy is used instead. Surface energyis measured in ergs per length squared, where one eng, the force of one dyne exertedfor a distance of one cm is equal to gram centimeter squared per second squared gcm2/s2 or, equivalently, 10−7 joules. The surface tension of water at room temperatureis γwater=72 dynes/cm, ethanol has a lower surface tension of γethanol=22 dynes/cm andmercury has a surface tension as large as γmercury=465 dynes/cm.

5.1.3 Cell membranes

The most intriguing of all biomembranes is the cell membrane, a semipermeable phos-pholipid bilayer common to all living cells. This lipid bilayer which is approximately 6nm thick consists of a variety of different biopolymers the most common of which areproteins, lipids and oligosaccharides.

phospholipid

water

cell membrane

waterFigure 5.7: Lipid bilayer of the cell membrane - characteristic arrangement of phospholipid moleculeswith hydrophilic polar head groups being oriented towards the aqueous phase while the hydrophobictails are oriented towards the non-polar inside

On the inside, the lipid bilayer serves as attachment for the cytoskeleton which is pri-marily responsible to controll the cell shape, see figure 5.7. On the outside, the cellmembrane plays an important role in attaching to the extracellular matrix. Specificproteins embedded in the cell membrane can act as molecular signals and to allow forcell to cell interaction. In funghi, bacteria and plants, the cell membrane is further sur-rounded by the cell wall. In an aqueous environment, the intact cell membrane seeks

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5 Biomembranes

cell membranephospholipid

bilayer

Figure 5.8: Lipid bilayer of cell membrane - characteristic arrangement of phospholipid molecules withhydrophilic polar head groups being oriented towards the aqueous phase while the hydrophobic tails areoriented towards the non-polar inside

to attain its lowest energy level. Accordingly, the nonpolar aminoacid residues of itsproteins and the fatty acid chains of its phospholipids will typically be sequestered fur-thest away from the aqueous solvent. The ionic and polar head groups of the proteins,the lipids and the oligosaccharides, in turn, will seek to be in contact with water, seefigure 5.8. Perhaps the most important lesson learned from the study of pure phospho-lipid bilayer membranes is that they spontaneously seal to form closed structures thatseparate two aqueous compartments. In the configuration of a plain sheet with ends inwhich the hydrophobic interior are in contact with water, bilayers are unstable. Theirtypical spherical architecture with no ends is the most stable state of a phospholipidbilayer.

5.2 Energy

From a structural mechanics point of view, biomembranes are characterized throughtheir very thin structure. As you have seen, the lipid bilayer of the cell membrane hasa thickness of approximately 6 nm. The typical dimensions of a cell are at least of theorder of µm. Therefore, it is quite common to treat biomembranes as shell structures. Ingeneral, the notion of shells is associated with thin, curved structures that are subjectedto loads that can cause in plane stretches and shear and out of plane bending. A specialcase of shells, a flat shell of zero curvature, would be referred to as a plate. Shells arestructural elements for which one dimension, the thickness, is much smaller than theirtwo other dimensions, the length and the width. Based on this dimensional restriction,specific kinematic assumptions can be made that significantly reduce and simplify theset of governing equations of three dimensional continua.

5.2.1 The Kirchhoff Love theory

The kinematic assumptions that seem reasonable for biomembranes are based on theclassical von Kármán theory. The von Kármán theory implies that the displacementsare small, while the rotations of the shell’s mid surface can be moderate. Of course,

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